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Full text of "Treatise On Analysis Vol-Ii"

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120 XII! INTEGRATION then a function u e Jf R(X) such that/- u £ 0, we may also assume that/^ 0. Now let N be the set of points x e X such that/(*) = + oo ; we have ncpN £f for all integers n > 0, hence >^*(<pN) £ /^*(/), and the hypothesis therefore implies that ju*(<pN) = 0. The relation "/(*) = #(*) almost everywhere in X " is an equivalence relation between mappings of X into a set E, because the union of two negligible sets is negligible. We say then that / and g are equivalent (with respect to u) or ^-equivalent, and we shall denote by / the equivalence class of a mapping/: X -> E. A mapping/: A -» E, where A is a subset of X, is said (by abuse of language) to be defined almost everywhere in X if X — A is negligible. The equivalence class of such a mapping /is then defined to be the equivalence class of any mapping of X into E which extends /; as before, we denote the equivalence class by /. Two functions / g defined almost everywhere are said to be equivalent if/ = g ; this means that the set of points x e X at which /and g are both defined and/(jc) = g(x) has a negligible set for its complement. (13.6.5) Let f, g be two equivalent mappings ofX into R. Then /x*(/) = H*(g)- Let N be the negligible set of points x e X such that /(jc) ^ g(x). Since the functions /, g and sup(/, g) are equal on X — N, we may assume that /<; g. Let h be the negligible function which is equal to + oo at the points of N, and to 0 elsewhere. If v e J is such that/^ v, the function v + h is defined at all points of X, and we have g :g v + h, so that by virtue of (13.5.6), because /**(#) > -co. From the definition of /**(/), it follows that ii*(g) ^ /**(/) and hence that /*%) = /**(/). If /is a mapping into R which is defined and finite almost everywhere in X, then /is equivalent to a finite function defined on the whole of X. If/ g are functions defined almost everywhere in X, with values in K, and finite almost everywhere, then the same is true of/+ g and/#, and the equivalence classes of these functions depend only on /and g. They are denoted byf+g and/*?, respectively. lff(x) ^ g(x) almost everywhere, then/^*) g g^x) almost everywhere for any functions /15 gl equivalent to /, g, respectively. In this case we write /<* #, and this defines an order relation on the set of equivalence classes (with respect to /*) of mappings of X into R.pact spaces and TT : X ->• Y a proper continuous mapping